How to build motivic realizations?

The theory of motives is a grand program envisioned by Grothendieck to encapsulate, within a single framework, the essential features shared by various cohomology theories developed by his school for smooth projective varieties over a field $k$, which are nowadays called Weil cohomology theories. Typical examples include $\ell$-adic cohomology, algebraic de Rham cohomology, and Betti cohomology. The notion of pure motives was introduced by Grothendieck, along with the expectation that there should exist a universal Weil cohomology theory reproducing all known properties of the existing ones. A natural candidate for the category of pure motives is the category of Chow motives introduced by Grothendieck. However, this approach immediately leads to the notorious standard conjectures, which remain unproven to this day. It is also natural to imagine that one can define motives for smooth but possibly non-projective varieties, thereby obtaining the notion of mixed motives $\mathrm{MM}(k)$, and then recover pure motives through other tools such as resolutions of singularities or semisimplifications. 

Furthermore, one may expect the existence of a relative version $\mathrm{MM}(X)$, where $X$ varies over varieties over $k$. Instead of directly searching for such a category, Deligne proposes to first construct its derived version $\mathbf{D}^b(\mathrm{MM}(X))$ and then recover $\mathrm{MM}(X)$ via a motivic t-structure. This idea is made precise by Beilinson and will be recalled below.

In a more modern formulation, this says that the category of constructible etale motives $\mathbf{DA}^{\text{et}}(X,\mathbb{Q})$ (originally due to Voevodsky but later developments attributed to Ayoub, Cisinski, Deglise) is the conjectural category $\mathbf{D}^b(\operatorname{MM}(X))$ and hence carries a motivic $t$-structure. At the level of abelian categories, the category $\operatorname{MM}(X)$ is initial in the sense that it could recover all known cohomology of smooth varieties. One may expect certain initiality of its (conjectural) derived version $\mathbf{DA}(X)$, which is what I want to show here in this post. 

Desire. How can one describe all derived categories in terms of etale motives $\mathbf{DA}^{\text{et}}(-,\mathbb{Q})$?  

Let me get started with some basic philosophy encoded in the following theorem of Drew and Gallauer using the language of $\infty$-coefficient system (at the level of triangulated categories, these are called stable homotopical functor by Ayoub and motivic theory by Cisinski + Deglise); although some of the core ideas are already back to Cisinski, Deglise. 

There is a technical detail that I have not yet mentioned. One has a similar construction for $\mathbf{DA}^{\text{et}}(X,\mathbb{Q})$, in fact, a finer construction, denoted by $\mathbf{SH}(X)$, using the Nisnevich topology (and has no $\mathbb{Q}$-linearity); both of them are shadows of some more general construction of Ayoub $\mathbf{SH}^{\tau}_{\mathfrak{M}}(X)$ where $\tau$ is a topology and $\mathfrak{M}$ is a nice category. In case $\tau = \text{et}$ and $\mathfrak{M} = \mathbf{Ch}(\mathbb{Q})$ chain complexes of $\mathbb{Q}$-vector spaces, one has $\mathbf{DA}^{\text{et}}(X,\mathbb{Q})$ and if $\tau = \text{Nis}$ and $\mathfrak{M} = \mathbf{Sptr}$ the category of $S^1$-spectra, then one has $\mathbf{SH}(X)$. In a word, $\mathbf{DA}^{\text{et}}(X,\mathbb{Q})$ is the etale $\mathbb{Q}$-linearity version of $\mathbf{SH}(X)$ and it can be represented as a certain category of module over the $\mathbb{Q}$-motivic cohomology spectrum.  

Theorem (Drew + Gaullauer). Let $k$ be a field, the collection $\mathbf{SH}(X)$ with $X$ a variety over $k$ forms a initial theory having a six-functor formalism. In other words, given another collection $\mathbf{D}(X)$ with six-functor formalism, there exists a realization functor

$$\mathbf{SH}(X) \longrightarrow \mathbf{D}(X)$$ commuting with six operations.

Behind the scene, one essentially needs the language of $\infty$-categories here. This causes no problem in practice since all known derived categories are $\infty$-categories in nature (if one wants to stay with model categories, one will need some more technical details). In particular, this (abstractly) recovers a theorem of Ayoub. 

Theorem (Ayoub). Let $k \subset \mathbb{C}$ be a subfield of complex, let $X$ be a $k$-variety and $X(\mathbb{C})$ be the set of $\mathbb{C}$-points endowed with the transcendental topology. There exists a Betti realization functor

$$\operatorname{Bti}^*_X \colon \mathbf{SH}(X) \longrightarrow \mathbf{D}(X(\mathbb{C}),\mathbb{Q})$$ where the RHS is the (unbounded) derived category with $\mathbb{Q}$-coefficients.

In fact, Ayoub proved something finer, there is a pair of Betti realizations on etale motives $$(\operatorname{Bti}^*_X \dashv \operatorname{Bti}_{*,X}) \colon \mathbf{DA}^{\text{et}}(X,\mathbb{Q}) \longrightarrow \mathbf{D}(X(\mathbb{C}),\mathbb{Q}).$$ This is a better one since one has $\mathbb{Q}$-linearity on both sides. Given the Betti realization, the Grothendieck's standard conjectures take the following modern form:

Theorem (Beilinson). The standard conjectures for fields $k \subset \mathbb{C}$ hold if and only if there exists a (motivic) $t$-structure on $\mathbf{DA}^{\text{et}}(k,\mathbb{Q})$ such that the Betti realization 

$$\operatorname{Bti}^*_k \colon  \mathbf{DA}^{\text{et}}_{\text{ct}}(k,\mathbb{Q}) \longrightarrow \mathbf{D}^b(\operatorname{Spec}(k)(\mathbb{C}),\mathbb{Q}) = \mathbf{D}^b(\mathbb{Q})$$ is $t$-exact, where the subscript ct means constructible motives and the superscript b means bounded and the RHS is equipped with the canonical $t$-structure. (I should stress that if standard conjectures hold for field, then motivic $t$-structure exists for varieries as well). 

Clearly, one should not stop at this point and might want to have a closer look at what's happening under the surface. 

Let us recall another construction of the RHS, step-by-step 

• Let $\mathrm{Sm}/X(\mathbb{C})$ be the category of (analytic) smooth $X(\mathbb{C})$-manifolds. 
• Let $\mathbf{Shv}_{\text{et}}(Sm/X(\mathbb{C}),\mathbb{Q})$ be the $\infty$-category of sheaves with respect to the analytic topology and with $\mathbb{Q}$-coefficient.
• Let $\mathbf{D}^1$ denote the closed unit disc, let $\mathbf{Shv}_{\text{et},\mathbf{D}^1}(Sm/X(\mathbb{C}),\mathbb{Q})$ be the full subcategory of $\mathbf{D}^1$-invariant sheaves, i.e., those $F$ with $F(U \times \mathbf{D}^1) \simeq F(U)$ ($U$ analytically smooth over $X(\mathbb{C})$) and obtain a category of effective analytic motives, $\mathbf{D}^{\text{eff}}_{\text{Betti}}(X)$. 
• Let $T = \operatorname{Coker}(\mathbb{Q} \to \mathbb{Q}(\mathbf{A}^1(\mathbb{C}) \setminus \left \{0 \right \}))$ (the map induced by unit section) be the (analytic) Lefschetz motive. We let $\mathbf{D}_{\text{Betti}}(X) = \mathbf{D}^{\text{eff}}_{\text{Betti}}(X)[T^{-1}]^{\otimes}$ be the stablization of  effective motives; namely, we make $T$ invertible with the (derived) tensor product. 

Theorem (Ayoub). There is a canonical equivalence $\mathbf{D}_{\text{Betti}}(X) \simeq \mathbf{D}(X(\mathbb{C}),\mathbb{Q})$. 

Thus we now want to understand $\operatorname{Bti}^* _X \colon  \mathbf{DA}^{\text{et}}(X,\mathbb{Q}) \longrightarrow \mathbf{D}_{\text{Betti}}(X)$. Let me first present the technical approach and then the (more interesting) philosophical approach. The technical one is to recall the definition of $\mathbf{DA}^{\text{et}}(X,\mathbb{Q})$ step-by-step:

• Let $Sm/X$ be the category of smooth (algebraic) $X$-varieties endowed with the etale topology. 
• Let $\mathbf{Shv}_{\text{et}}(Sm/X,\mathbb{Q})$ be the $\infty$-category of sheaves with respect to the etale topology and with $\mathbb{Q}$-coefficient.
• Let $\mathbb{A}^1_X$ denote the affine line, let $\mathbf{Shv}_{\text{et},\mathbb{A}^1}(Sm/X,\mathbb{Q})$ be the full subcategory of $\mathbb{A}^1$-invariant sheaves, i.e., those $F$ with $F(U \times \mathbb{A}^1) \simeq F(U)$ ($U$ smooth over $X$) and obtain a category of effective etale motives, $\mathbf{DA}^{\text{et,eff}}(X,\mathbb{Q})$. 
•  Let $T = \operatorname{Coker}(\mathbb{Q} \to \mathbb{Q}(\mathbb{G}_{m,X})$ be the Lefschetz motive, where the map inside bracket is induced by the unit section. The stabilization $\mathbf{DA}^{\text{et}}(X,\mathbb{Q}) = \mathbf{DA}^{\text{et,eff}}(X,\mathbb{Q})[T^{-1}]^{\otimes}$ is the desired one. 

 At this point, one can guess how we build the Betti realization. We have a natural morphism $Sm/X \to Sm/X(\mathbb{C})$ and by carefully checking step-by-step of constructions on both sides. We end up with the desire realization. So this is not really a wow for us, given Ayoub's theorem above. The interesting one is the following: the object $\mathcal{B}_X = \operatorname{Bti}_{*,X}(\mathbb{Q})$ is naturally a commutative algebra object, called the Betti algebra.

It is hard to overstate the importance of this algebra. In fact, I feel an urge to state the following theorem by Ayoub, though not relevant to what I want to present here.

Theorem (Ayoub and Gallauer, Choudhurry). The object $\operatorname{Bti}^*_k(\mathcal{B}_k) \in \mathbf{D}(\mathbb{Q})$ has no positive cohomology and it is an actual Hopf $\mathbb{Q}$-algebra whose spectrum is the motivic Galois group of Nori $\mathcal{G}^{\text{Nori}}(k)$ (note that conjecturally, $\operatorname{Rep}(\mathcal{G}^{\text{Nori}}(k)) \simeq \operatorname{MM}(k)$). 

Let us come back to our main interest.  

Theorem (Cisinski, Deglise). One has a canonical equivalence

$$\mathbf{D}(X(\mathbb{C}),\mathbb{Q}) \simeq \operatorname{Mod}_{\mathcal{B}_X}(\mathbf{DA}^{\text{et}}(X,\mathbb{Q})),$$ where the RHS is the $\infty$-category of $\mathcal{B}_X$-modules. 

This leads to the following idea

Slogan (Cisinski, Deglise). All realization from the category of etale motives $\mathbf{DA}^{\text{et}}(-,\mathbb{Q})$ arise this way, namely, as certain category of modules over some commutative algebra in $\mathbf{DA}^{\text{et}}(-,\mathbb{Q})$.

This holds true for all known existing theories such as mixed Hodge modules, $\ell$-adic sheaves and recently Nori motives (by Swann Tubach). How can we describe this? We observe that the Betti realization induces a functor on modules

$$\operatorname{Mod}_{\mathcal{B}_X}(\mathbf{DA}^{\text{et}}(X,\mathbb{Q})) \longrightarrow \operatorname{Mod}_{\mathbb{Q}}(\mathbf{D}(X(\mathbb{C}),\mathbb{Q})) \simeq \mathbf{D}(X(\mathbb{C}),\mathbb{Q})$$ (this is a general fact, and has nothing to do with motives) and combining with the "free" functor 

$$\mathbf{DA}^{\text{et}}(X,\mathbb{Q}) \overset{ \mathcal{B}_X \otimes (-)}{\longrightarrow} \operatorname{Mod}_{\mathcal{B}_X}(\mathbf{DA}^{\text{et}}(X,\mathbb{Q}))$$ we recover the Betti realization.  

Theorem. (Cisinski, Deglise) The functor  

$$\widetilde{\operatorname{Bti}}^*_X \colon \operatorname{Mod}_{\mathcal{B}_X}(\mathbf{DA}^{\text{et}}(X,\mathbb{Q})) \longrightarrow  \mathbf{D}(X(\mathbb{C}),\mathbb{Q})$$ is an equivalence of categories.

Sketch of proof. Clearly, we want to show that $\mathrm{id} \longrightarrow \widetilde{\operatorname{Bti}}_{*,X}\widetilde{\operatorname{Bti}}^*_X$ where $\widetilde{\operatorname{Bti}}_{*,X}$ is its right adjoint (existence guaranteed by adjoint functor theorem on presentable $\infty$-categories). The left adjoint $\widetilde{\operatorname{Bti}}_X^*$ preserves compact objects and hence the right adjoint $\widetilde{\operatorname{Bti}}_{*,X}$ preserves (homotopy) colimits. It is sufficient to show that the counit 

$$C \longrightarrow \widetilde{\operatorname{Bti}}_{*,X}\widetilde{\operatorname{Bti}}^*_X(C)$$ is an equivalence with $C$ a compact object. Now the forgetful functor $\operatorname{Mod}_{\mathcal{B}_X}(\mathbf{DA}^{\text{et}}(X,\mathbb{Q})) \longrightarrow \mathbf{DA}^{\text{et}}(X,\mathbb{Q})$ is conservative, it is enough to check the above on $\mathbf{DA}^{\text{et}}(X,\mathbb{Q})$ $$C \longrightarrow \operatorname{Bti}_{*,X}\operatorname{Bti}^*_X(C).$$ Now the case $X = \operatorname{Spec}(k)$ is true thanks to the technical lemma below, since compact objects are dualisable. The case of general variety $X$ is reduced to the case of $\operatorname{Spec}(k)$ by using a formal adjoint argument on six operations. The functor is essentially surjective because formally they are compactly generated by motives of smooth varieties. 

Lemma. Let $(f \dashv g) \colon \mathcal{M} \longrightarrow \mathcal{N}$ be a pair of adjoint functor between unital monoidal categories. Suppose that $f$ is unital monoidal (so $g$ is lax monoidal). In order to have the projection map $g(A) \otimes B \longrightarrow g(A \otimes f(B))$ to be an isomorphism, it suffices to assume that $B$ is strongly dualisable. 

It is now worth mentioning a recent result by Tubach on Nori motives. The long story comes back to Nori who defined Nori motives over a field $k \subset \mathbb{C}$ as a candidate for $\mathrm{MM}(k)$ mixed motives over $k$. It is Florian Ivorra and Sophie Morel came up with the idea that the relative version $\mathrm{MM}(X)$ ($X$ is a $k$-variety) behaves like a category of perverse sheaves. This leads to the notion of perverse Nori motives (and it is Arapura who thinks about $\mathrm{MM}(X)$ as a category of constructible sheaves; clearly, over a point, they coincide). Currently, Nori motives is a rich source to study motives after Ivorra, Morel. The definition of Nori motives is fairly simple:

Definition. Let $k \subset \mathbb{Q}$ and $X/k$ a variety, the abelian $\mathbb{Q}$-linear category of perverse Nori motives $\operatorname{MPerv}(X)$ is defined as the universal abelian factorization of the following composition

$$\mathbf{DA}^{\text{et}}_{\text{ct}}(X,\mathbb{Q}) \overset{\operatorname{Bti}^*_X}{\longrightarrow} \mathbf{D}^b_{\text{ct}}(X,\mathbb{Q}) \overset{{}^p\mathrm{H}^0}{\longrightarrow} \operatorname{Perv}(X),$$ where $\operatorname{Perv}(X)$ denotes the category of perverse sheaves in the usual sense. What does it mean to be a universal abelian factorization? Well, this means that the above composition factors as

$$\mathbf{DA}^{\text{et}}_{\text{ct}}(X,\mathbb{Q}) \overset{h_X}{\longrightarrow} \operatorname{MPerv}(X) \overset{\operatorname{rat}_X}{\longrightarrow} \operatorname{Perv}(X),$$ where $h_X$ is a cohomological functor and $\operatorname{rat}_X$ is exact, faithful and this decomposition is initial among all decompositions of the same form. The existence of such a decomposition is dated back to Freyd, Neeman, etc. 

Why is this definition a "correct" one and what can we expect from this?  

• Historically, the definition for $X= \operatorname{Spec}(k)$ is actually a theorem of Nori. Nori first defined $\operatorname{MPerv}(k)$ as a certain category of quiver representations over pairs $(X(\mathbb{C}),Z(\mathbb{C}),n \in  \mathbb{Z})$ (where $Z \subset X$ closed), and proves that this definition is an universal abelian factorization. Thus, Ivorra, Morel's definition turns Nori's theorem into a definition.
• The functor $\mathbf{DA}^{\text{et}}_{\text{ct}}(X,\mathbb{Q}) \overset{h_X}{\longrightarrow} \operatorname{MPerv}(X)$ should be understood that the zero-th motivic cohomological functor and since motives are expected to be universal, the above definition is reasonable to certain extent.
• As we discuss at the beginning, one should have that $\mathbf{DA}^{\text{et}}_{\text{ct}}(X,\mathbb{Q}) \simeq \mathbf{D}^b(\operatorname{MPerv}(X))$ (one can think of this as a motivic version of Beilinson's theorem $\mathbf{D}^b_{\text{ct}}(X(\mathbb{C}),\mathbb{Q}) \simeq \mathbf{D}^b(\operatorname{Perv}(X))$. 
• If the above holds true, then $$\operatorname{rat}_X \colon \mathbf{D}^b_{\text{ct}}(X(\mathbb{C}),\mathbb{Q})  \simeq \mathbf{D}^b(\operatorname{MPerv}(X)) \longrightarrow \mathbf{D}^b(\operatorname{Perv}(X)) \simeq \mathbf{D}^b_{\text{ct}}(X(\mathbb{C}),\mathbb{Q})$$ should be preicisely the Betti realization. 
• Can we build six-functor formalism for $\mathbf{D}^b(\operatorname{MPerv}(X))$ as it should have?
• Can we recover the Betti realization from $\operatorname{rat}_X$? Without refering to standard conjectures? 

It is interesting that the two last questions have positive answers!

Theorem (Ivorra, Morel and Terenzi). The collection $\mathbf{D}^b(\operatorname{MPerv}(X))$ is encoded with a six-functor formalism and a weight theory similar to the one of $\ell$-adic sheaves.  

At this point, everything exists in the world of triangulated categories. To use tools such as Drew, Gallauer's theorem, we should be able to upgrade $\mathbf{D}^b(\operatorname{MPerv}(X))$ to $\infty$-categories. 

Theorem (Tubach). The collection $\mathbf{D}^b(\operatorname{MPerv}(X))$ and their six functors naturally live in the world of $\infty$-categories and $\infty$-functors. 

Now by plugging the Drew, Gallauer's theorem in, we derive that there exists a realization functor

$$\mathbf{SH}(X) \longrightarrow \operatorname{Ind}(\mathbf{D}^b(\operatorname{MPerv}(X))),$$ where $\operatorname{Ind}$ indicates the $\infty$-ind-completion (this is also a technical reason why we want $\infty$-category, there is no such thing for triangulated categories - the ordinary ind-completion is not the correct one here). But as the case for Betti realization, the RHS $\operatorname{Ind}(\mathbf{D}^b(\operatorname{MPerv}(X)))$ is naturally $\mathbb{Q}$-linear and hence (after some non-trivial arguments), one can show that the above one factors through a Nori realization 

$$\operatorname{Nri}^*_X \colon \mathbf{DA}^{\text{et}}(X,\mathbb{Q}) \longrightarrow \operatorname{Ind}(\mathbf{D}^b(\operatorname{MPerv}(X)))$$ commuting with six operations. Now we are at the point analogous to Betti realizations. We have the the Nori algebra $\mathcal{N}_X = \operatorname{Nri}_{*,X}(\mathbf{1}_X)$, where $\operatorname{Nri}_*$ is the right adjoint of $\operatorname{Nri}^*$ and $\mathbf{1}_X$ is the unit of the monoidal structure on $\mathbf{D}^b(\operatorname{MPerv}(X))$. We have the following analogue of Cisinski, Deligse.

Theorem (Tubach and Emil, Terenzi). The following statements hold true:

• There is a canonical equivalence of $\infty$-categories $$\operatorname{Ind}(\mathbf{D}^b(\operatorname{MPerv}(X))) \simeq \operatorname{Mod}_{\mathcal{N}_X}(\mathbf{DA}^{\text{et}}(X,\mathbb{Q}))$$ such that the Nori realization is given by $$\mathbf{DA}^{\text{et}}(X,\mathbb{Q}) \overset{\mathcal{N}_X \otimes (-)}{\longrightarrow}  \operatorname{Mod}_{\mathcal{N}_X}(\mathbf{DA}^{\text{et}}(X,\mathbb{Q})) \simeq  \operatorname{Ind}(\mathbf{D}^b(\operatorname{MPerv}(X))),$$ where the first map is as usual the free functor.
• The composition $\mathbf{DA}^{\text{et}}_{\text{ct}}(X,\mathbb{Q}) \overset{\operatorname{Nri}^*_X}{\longrightarrow} \mathbf{D}^b(\operatorname{MPerv}(X)) \overset{\mathrm{H}^0}{\longrightarrow} \operatorname{MPerv}(X)$ is exactly $h_X$, the universal zero-th cohomological functor. 
• The composition $\mathbf{DA}^{\text{et}}_{\text{ct}}(X,\mathbb{Q}) \overset{\operatorname{Nri}^*_X}{\longrightarrow} \mathbf{D}^b(\operatorname{MPerv}(X)) \overset{\operatorname{rat}_X}{\longrightarrow} \mathbf{D}^b(\operatorname{Perv}(X)) \simeq \mathbf{D}^b_{\text{ct}}(X(\mathbb{C}),\mathbb{Q})$ is nothing but the Betti realization $\operatorname{Bti}^*_X$. 

All of this in some sense is a miracle, one defines something in a purely abstract way (as a universal abelian factorization), then derives six operations and weights and again gets a Nori realization via modules and gets back the purely abstract definition and even the Betti realization  once again via the Nori realization. The land of Nori motives is still huge and rich for applications that I hope it will reach closer to applications in other branches of geometry.